SMP #2 Reason Abstractly and Quantitatively

The North Carolina Unpacked Standards provide a summary statement about the SMPs for each grade level:Kindergarten: Mathematically proficient students in Kindergarten make sense of quantities and the relationships while solving tasks. This involves two processes- decontexualizing and contextualizing. In Kindergarten, students represent situations by decontextualizing tasks into numbers and symbols. For example, in the task, “There are 7 children on the playground and some children go line up. If there are 4 children still playing, how many children lined up?” Kindergarten students are expected to translate that situation into the equation: 7-4 = ___, and then solve the task. Students also contextualize situations during the problem solving process. For example, while solving the task above, students refer to the context of the task to determine that they need to subtract 4 since the number of children on the playground is the total number of students except for the 4 that are still playing. Abstract reasoning also occurs when students measure and compare the lengths of objects.

1st Grade: Mathematically proficient students in Grade 1 make sense of quantities and the relationships while solving tasks. This involves two processes- decontexualizing and contextualizing. In Grade 1, students represent situations by decontextualizing tasks into numbers and symbols. For example, in the task, “There are 60 children on the playground and some children go line up. If there are 20 children still playing, how many children lined up?” Grade 1 students are expected to translate that situation into the equation: 60 – 20 = ___ and then solve the task. Students also contextualize situations during the problem solving process. For example, while solving the task above, students refer to the context of the task to determine that they need to subtract 20 since the number of children on the playground is the total number except for the 20 that are still playing. The processes of reasoning also applies to Grade 1, as they look at ways to partition 2-dimensional geometric figures into halves, and fourths.

2nd Grade: Mathematically proficient students in Grade 2 make sense of quantities and the relationships while solving tasks. This involves two processes- decontexualizing and contextualizing. In Grade 2, students represent situations by decontextualizing tasks into numbers and symbols. For example, in the task, “There are 25 children in the cafeteria and they are joined by 17 more children. Then, if 19 of those children then leave, how many are still there?” Grade 2 students are expected to translate that situation into the equation: 25 + 17 – 19 = __ and then solve the task. Students also contextualize situations during the problem solving process. For example, while solving the task above, students can refer to the context of the task to determine that they need to subtract 19 since 19 children leave. The processes of reasoning also apply to Grade 2 as students begin to measure with standard measurement units by determining the length of quantities based on particular units of measure.

3rd Grade: Mathematically proficient students in Grade 3 recognize that a number represents a specific quantity. They connect the quantity to written symbols and create a logical representation of the problem at hand, considering both the appropriate units involved and the meaning of quantities. This involves two processes- decontexualizing and contextualizing. In Grade 3, students represent situations by decontextualizing tasks into numbers and symbols. For example, in the task, “There are 8 bags of cookies with the same amount of cookies in each bag. If there are 48 cookies how many cookies are in each bag?” Grade 3 students are expected to translate that situation into the equation: 8 * __ = 48 or 48 / 8 = __ and then solve the task. Students also contextualize situations during the problem solving process. For example, while solving the task above, students can refer to the context of the task to determine that they were given the number of bags, and the total number of cookies, but they need to find the number of cookies in each bag.

4th Grade: Mathematically proficient students in Grade 4 recognize that a number represents a specific quantity. They extend this understanding from whole numbers to their work with fractions and decimals. This involves two processes- decontexualizing and contextualizing. Grade 4 students decontextualize by taking a real-world problem and writing and solving equations based on the word problem. For example, consider the task, “if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Students will decontextualize by writing the equation 3/8 * 5 or repeatedly add 3/8 5 times. While students are working they will contextualize their work- knowing that the answer 15/8 or 1 7/8 represents the total number of pounds of roast beef that will be needed. Further, Grade 4 students write simple expressions, record calculations with numbers, and represent or round numbers using place value concepts.

5th Grade: Mathematically proficient students in Grade 5 recognize that a number represents a specific quantity. They extend this understanding from whole numbers to their work with fractions and decimals. This involves two processes- decontexualizing and contextualizing. Grade 5 students decontextualize by taking a real-world problem and writing and solving equations based on the word problem. For example, consider the task, “There are 2 2/3 of a yard of rope in the shed. If a total of 4 1/6 yard is needed for a project, how much more rope is needed?” Students decontextualize the problem by writing the equation 4 1/6 – 2 2/3 = ___ and then solving it. Further, students contextualize the problem after they find the answer, by reasoning that 1 3/6 or 1 ½ yards of rope is the amount needed. Further, Grade 4 students write simple expressions that record calculations with numbers and represent or round numbers using place value concepts.

InsideMathematics has video examples of SMP #2 in lessons from 5th grade classrooms and there is another video example below.

Noristown Unified School District in Pennsylvania has published some outstanding newsletters which focus on each of the Standards for Mathematical Practice. SMP Newsletter #2 can be found here.